Q: You meet a stranger on the street, and ask how many children he has. He
truthfully says two. You ask "Is the older one a girl?" He truthfully
says yes. What is the probability that both children are girls? What
would the probability be if your second question had been "Is at least
one of them a girl?", with the other conditions unchanged?

S: There are four equally-likely possibilities

Oldest child Youngest child

1. Girl Girl
2. Girl Boy
3. Boy Girl
4. Boy Boy

If the stranger says "My oldest child is a girl," he has eliminated
cases 3 and 4, and in the remaining cases both are girls 1/2 of the
time. If the stranger says "At least one of my children is a girl," he
has eliminated case 4 only, and in the remaining cases both are girls
1/3 of the time.

In actuality, there are about 106 boys born for every 100 girls, and
the odds of having a second child are influenced by the sex of the
first, so these four possibilities are not really equally likely.
According to a reader of the "Ask Marilyn" column (Parade magazine,
October 19, 1997), the US Census Bureau surveyed 42,888 two-children
families between 1987 and 1993. Of these, 11,334 (26.4%) were boy-boy,
11,118 (25.9%) were boy-girl, 10,913 (25.4%) were girl-boy and 9523
(22.2%) were girl-girl. Using these figures, if the oldest child
is a girl, then the odds that they both are girls is 46.6%, whereas if
at least one is a girl, the odds that they both are girls is 30.2%.